Lastly we want to comment on the evolution from the spin liquid in the \(x=0\)  line to the Z\(_2\)  SL I + PSF phase in the Fig. REF . Right now there are debates on whether this spin liquid in the intermediate \(\frac{J^\prime _b}{J_b}\)  at the \(x=0\)  limit is in a Z\(_2\)  spin liquid[1]}, [2]}, [3]}, [4]} or in a U(1) Dirac spin liquid[5]}, [6]}. If it is a Z\(_2\)  spin liquid, then it should be described by the zero flux ansatz of the Schwinger boson theory because it has short ranged \(120^\circ \)  correlations. In this case, doping the exciton induces a chemical potential tuned insulator to superfluid phase transition for the paired excitons on top of the Z2 spin liquid which leads to the Z\(_2\)  SL I + PSF phase found in our numerical calculation. If the layer polarized spin liquid is a U(1) Dirac spin liquid, then we should use fermionic spinons such as in Eq. REF  to describe the spin liquid phase. Then doping excitons will create spinon Fermi surfaces for both layers. But there is no signature of spinon Fermi surface in our numerical results at \(x=\frac{1}{18}\) . To reach the Z\(_2\)  SL I + PSF phase, we need the fermion spinons to further have pairing terms \(\epsilon _{\sigma \sigma ^{\prime }} f_{a;\sigma }f_{a;\sigma ^{\prime }}\)  for both layer \(a=t,b\) , which higgses the U(1) gauge field down to Z\(_2\) . Because this is a \(\pi \)  flux ansatz of fermionic spinon, it is equivalent to the zero flux phase of the Schwinger bosonIn this \(Z_2\)  spin liquid, the bosonic \(e\)  particle is a bound state of fermionic spinon and the vison. The vison feels a \(\pi \)  flux and hence the \(\pi \)  flux of fermionic spinon corresponds to the zero-flux of the bosonic spinon. and is in the same Z2 SL I phase. Thus we see that there need to be two phase transitions starting from the U(1) Dirac spin liquid phase to reach the Z\(_2\)  SL I +PSF phase. Future calculations at infinitesimal \(x\)  regime may be able to resolve the debates between the Z\(_2\)  spin liquid and Dirac spin liquid in the \(x=0\)  limit.
